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Chapter 22: Ratio, Proportion and Proof

Ratios

  • Ratios are used to compare quantities

  • You can find equivalent ratios by multiplying or dividing by the same number

Simplest form

  • To write a ratio in its simplest form, find an equivalent ratio with the smallest possible whole number values

Problem solved

  • If you use x to represent one part of the ratio, then you can write the length of the x as x

  • Then form an equation and solve it to find x

Harder questions

  • If the ratio contains decimals or fractions-multiply

  • If the ratio has mixed units-convert to the smaller unit

Proportion

  • Two quantities are in direct proportion when both quantities increase at the same rate

  • Two quantities are in inverse proportion when one quantity increases at the same rate as the other quantity decreases

Divide or multiply

  • You can use common sense to work out whether to divide or multiply in proportion questions

    • 6 people build a wall in 4 days

    • 6x4=24 so 1 person could build the wall in 24 days

    • 24/8=3 so 8 people could build the wall in 3 days

Examiners report

  • Show your method by writing down what you are working out at each stage of your working

  • Read the question carefully-you have to write down the option which offers better value

  • You won’t get the final mark if you just circle or underline your choice

Proof

  • You can use algebra to prove facts about numbers

  • Using algebra helps you to probe that something is true for every number

  • In a proof question the working is the answer

  • If the question say ‘show that’ or ‘prove that’ you need to write down every stage of your working

    • If you don’t you won’t get all the marks

Golden rule

  • If you need to prove something about numbers then you always use algebra

Algebraic proof toolkit

  • Even number = 2n

  • Odd number = 2n+1 or 2n-1

  • Multiple of 3 = 3n

  • Consecutive numbers = n,n+1.n+2…

  • Consecutive even numbers = 2n,2n+2,2n+4…

  • Consecutive odd numbers = 2n+1,2n+3,2n+5…

Chapter 22: Ratio, Proportion and Proof

Ratios

  • Ratios are used to compare quantities

  • You can find equivalent ratios by multiplying or dividing by the same number

Simplest form

  • To write a ratio in its simplest form, find an equivalent ratio with the smallest possible whole number values

Problem solved

  • If you use x to represent one part of the ratio, then you can write the length of the x as x

  • Then form an equation and solve it to find x

Harder questions

  • If the ratio contains decimals or fractions-multiply

  • If the ratio has mixed units-convert to the smaller unit

Proportion

  • Two quantities are in direct proportion when both quantities increase at the same rate

  • Two quantities are in inverse proportion when one quantity increases at the same rate as the other quantity decreases

Divide or multiply

  • You can use common sense to work out whether to divide or multiply in proportion questions

    • 6 people build a wall in 4 days

    • 6x4=24 so 1 person could build the wall in 24 days

    • 24/8=3 so 8 people could build the wall in 3 days

Examiners report

  • Show your method by writing down what you are working out at each stage of your working

  • Read the question carefully-you have to write down the option which offers better value

  • You won’t get the final mark if you just circle or underline your choice

Proof

  • You can use algebra to prove facts about numbers

  • Using algebra helps you to probe that something is true for every number

  • In a proof question the working is the answer

  • If the question say ‘show that’ or ‘prove that’ you need to write down every stage of your working

    • If you don’t you won’t get all the marks

Golden rule

  • If you need to prove something about numbers then you always use algebra

Algebraic proof toolkit

  • Even number = 2n

  • Odd number = 2n+1 or 2n-1

  • Multiple of 3 = 3n

  • Consecutive numbers = n,n+1.n+2…

  • Consecutive even numbers = 2n,2n+2,2n+4…

  • Consecutive odd numbers = 2n+1,2n+3,2n+5…